Any rectangular arrangement of numbers in m rows and n columns is called a matrix of order m×n. Matrices and determinants is an important topic for the JEE exam. These formulas will help students to have a quick revision before the exam. Students can expect 2-3 questions from this topic.
Matrices and Determinants Formulas
Any rectangular arrangement of numbers in m rows and n columns is called a matrix of order m×n.
Where aij denotes the element of the ith row and jth column. The above matrix is denoted as [aij]m×n . The elements a11, a22, a33 etc are called diagonal elements. Their sum is called the trace of A denoted by Tr(A).
2. Basic Definitions
(i) Row matrix: A matrix having one row is called a row matrix.
(ii) Column matrix: A matrix having one column is called a column matrix.
(iii) Square matrix: A matrix of order m×n is called square matrix if m = n.
(iv) Zero matrix: A = [aij]m×n is called a zero matrix, if aij = 0 for all i and j.
(v) Upper triangular matrix: A = [aij]m×n is said to be upper triangular, if aij= 0 for i > j.
(vi) Lower triangular matrix: A = [aij]m×n is said to be lower triangular, if aij = 0 for i < j.
(vii) Diagonal matrix: A square matrix [aij]m×n is said to be diagonal, if aij = 0 for i ≠ j.
(viii) Scalar matrix: A diagonal matrix A = [aij]m×n is said to be scalar, if aij = k for i = j.
(ix) Unit matrix (Identity matrix): A diagonal matrix A = [aij]n is a unit matrix, if aij = 1 for i = j.
(x) Comparable matrices: Two matrices A and B are comparable, if they have the same order.
3. Equality of matrices: Two matrices A = [aij]m×n and B = [bij]p×q are are said to be equal, if m = p and n = q and aij = bij ∀ i and j.
4. Multiplication of a matrix by a scalar: Let λ be a scalar, then λA = [bij]m×n where bij= λaij ∀ i and j.
5. Addition of matrices: Let A = [aij]m×n and B = [bij]m×n be two matrices, then A+B = [aij]m×n+ [bij]m×n = [cij]m×n where cij = aij+bij ∀ i and j.
6. Subtraction of matrices: A-B = A+(-B), where -B = ( -1)B.
7. Properties of addition and scalar multiplication:
(i) λ(A+B) = λA+λB
(ii) λA = Aλ
(iii) (λ1+λ2)A = λ1A+λ2A
8. Multiplication of matrices: Let A = [aij]m×p and B = [bij]p×n , then AB = [cij]m×n where cij =
9. Properties of matrix multiplication:
(i) AB ≠ BA
(ii) (AB)C = A(BC)
(iii) AIn = A = InA
(iv) For every non singular square matrix A (i.e., | A |≠ 0 ) there exists a unique matrix B so that AB = In = BA. In this case we say that A and B are multiplicative inverses of one another. I.e., B = A-1 or A = B-1 .
10. Transpose of a Matrix.
Let A = [aij]m×n then A’ or AT the transpose of A is defined as A’ = [aji]n×m .
(i) (A’)’ = A
(ii) (λA)’ = λA’
(iii) (A+B)’ = A’+B’
(iv) (A-B)’ = A’-B’
(v) (AB)’ = A’B’
(vi) For a square matrix A, if A’ = A , then A is said to be a symmetric matrix.
(vii) For a square matrix A, if A’ = -A , then A is said to be a skew symmetric matrix.
11. Submatrix of a matrix:
Let A be a given matrix. The matrix obtained by deleting some rows and columns of A is called a submatrix of A.
12. Properties of determinant:
(i) | A | = | A’ | for any square matrix A.
(ii) If two rows or two columns are identical, then | A | = 0.
(iii) If | λA | = λn| A |, when A = [aij]n×n.
(iv) If A and B are two square matrices of the same order, then | AB |= | A || B |
13. Singular and Non-singular matrix:
A square matrix A is said to be singular, if | A | = 0.
A square matrix A is said to be non-singular, if | A | ≠ 0.
14. Cofactor and adjoint matrix.
Let A = [aij]n×n be a square matrix. The matrix obtained by replacing each element of A by the corresponding cofactor is called the cofactor matrix of A. The transpose of the matrix of the cofactor of A is called the adjoint of A, denoted as adj A.
15. Properties of adj A.
(i) A . adj A = | A |In = (adj A)A where A = [aij]n×n.
(ii) | adj A | = | A |n-1 , where n is the order of A.
(iii) If A is a symmetric matrix, then adj A is also symmetric.
(iv) If A is singular, then adj A is also singular.
(v) Let A be non singular matrix, then
(vi) (A-1)T = (AT)-1 for any non singular matrix.
(vii) (A-1)-1 = A, if A is non singular.
(viii) A-1 is always non singular.
(ix) (adj AT ) = (adj A)T
(x) Let k be a non zero scalar and A be a non singular matrix, then (kA)-1 =
(xi) | A-1 | =
(xii) Let A be a non singular matrix, then AB = AC⇒ B = C and BA = CA ⇒ B = C.
16. System of linear equations and matrices:
System of linear equations AX = B is said to be consistent if it has at least one solution.
(i) System of linear equations and matrix inverse:
(a) If A is non-singular, solution is given by X = A-1B.
(b) If A is singular, adj(A) B = 0 and no two columns of A are proportional, then the system has infinitely many solutions.
(c) If A is singular and (adj A)B ≠ 0, then the system has no solution.
(ii) Homogeneous system and matrix inverse:
If the above system is homogeneous, n equations in n unknowns, then in the matrix form it is AX = 0. ( since b1 = b2 =….. bn = 0), where A is a square matrix.
If A is non-singular, the system has only one trivial solution. X = 0.
If A is singular, then the system has infinitely many solutions (including the trivial solution) and hence it has non-trivial solutions.
(iii) Elementary row transformation of Matrix:
The following operations on a matrix are called elementary row transformations.
(a) Interchanging two rows.
(b) Multiplication of all the elements of a row by a non zero scalar.
(c) Addition of a constant multiple of a row to another row.
17. Characteristic Polynomial and characteristic equation.
Let A be a square matrix, then the polynomial | A-xI | is called the characteristic polynomial of A and the equation | A-xI | = 0 is called the characteristic equation of A.
18. Cayley Hamilton theorem:
Every square matrix A satisfies its characteristic equation. I.e., a0xn+a1xn-1+…..+an-1x+an = 0 is the characteristic equation of A, then a0An+a1An-1+……..+an-1A+anI = 0
19. More definitions on matrices:
(i) Nilpotent matrix: A square matrix A is called nilpotent if Ap = 0 for some positive integer. If p is the smallest such positive integer, then p is called its nilpotency.
(ii) Idempotent matrix: A square matrix A is said to be idempotent if, A2 = A.
(iii) involutory matrix: A square matrix A is said to be involutory if, A2 = I.
(iv) Orthogonal matrix: A square matrix A is said to be orthogonal if, ATA = I = AAT.
(v) Unitary matrix: A square matrix A is said to be unitary if
We write the expression a1b2-a2b1 as
1. Expansion of determinant:
D = a1(b2c3 – b3c2) – b1(a2c3 – a3c2) + c1(a2b3 – a3b2)
2. Minors: The minor of aij is obtained by deleting ith row and jth column from the determinant. It is denoted by Mij.
3. Cofactor: Cofactor of element aij is Cij = (-1)i+j Mij
D = a11M11 – a12M12+a13M13 = a11C11+a12C12+a13C13
4. Transpose of a Determinant: The transpose of a determinant is a determinant obtained after interchanging the rows and columns.
5. Symmetric, Skew symmetric, Asymmetric Determinants:
(i) A determinant is symmetric if it is identical to its transpose. The ith row is identical to its ith column. I.e. aij = aji for all values of i and j.
(ii) A determinant is skew symmetric if it is identical to its transpose having the sign of each element inverted. I.e. aij = -aji for all values of i and j.
(iii) A determinant is asymmetric if it is neither symmetric nor skew symmetric.
6. Properties of determinants:
(i) D = D’
(ii) If a determinant has all the elements zero in any row or column, then D = 0
(iii) If any two rows or columns of a determinant be interchanged, then D’ = -D.
(iv) If a determinant has any two rows or columns identical, then D = 0.
(v) If all the elements of any row or column be multiplied by the same number k, then D’ = kD.
(vi) If each element of any row or column can be expressed as a sum of two terms, then the determinant can be expressed as the sum of two determinants. i.e.
(vii) The value of a determinant is not altered by adding to the elements of any row or column a constant multiple of the corresponding elements of any other row or column.
7. Multiplication of two determinants:
8. Summation of determinants:
Let Δ(r) =
Where a1, a2, a3, b1, b2, b3 are constants independent of r.
9. Integration of a determinant:
10. Differentiation of determinant:
11. Cramer’s rule:
(i) Two variables:
(a) Consistent equations: Definite and unique solution. [Intersecting lines]
(b) Inconsistent equation: No solution. [Parallel line]
(c) Dependent equation: Infinite solutions. [Identical lines]
Let a1x+b1y+c1 = 0 and a2x+b2y+c2 = 0 then,
(ii) Three variables:
Let a1x+b1y+c1z = d1 , a2x+b2y+c2z = d2 and a3x+b3y+c3z = d3, then x= D1/D, y = D2/D and z = D3/D, where D =
(iii) Consistency of a system of equations:
(a) If D ≠ 0 and at least one of D1, D2, D3 ≠ 0, then the given system of equations are consistent and has a unique non trivial solution.
(b) If D ≠ 0 and D1 = D2 = D3 = 0, then the given system of equations are consistent and has trivial solution only.
(c) If D = D1 = D2 = D3 = 0, then the given system of equations have either infinite solutions or no solution.
(d) If D = 0 but atleast one of D1, D2, D3 is not zero then the equations are inconsistent and have no solution.
(e) If a given system of linear equations has only zero solution for all its variables, then the given equations are said to have trivial solution.
(iv) Three equation in two variable:
If x and y are not zero, then the condition for a1x+b1y+c1 = 0; a2x+b2y+c2 = 0 and a3x+b3y+c3 = 0 to be consistent in x and y is
13. Application of Determinants:
(i) Area of a triangle whose vertices are (xr, yr) ;r = 1,2,3 is D =